The representation theory of quantum groups including at roots of unity is an important part of Lie representation theory. In this talk, we will study one of categories of representations: the quantum category 𝒪, which is a suitable analogue of the classical Bernstein-Gelfand category 𝒪. We will relate it to a model representation category, the affine Hecke category, more precisely to the heart of the new t-structure on that category (all these terms will be defined in the lectures).
To study topological Hochschild homology as an invariant of stable ∞-categories and endow it with its universal property in this context, Nikolaus introduced the formalism of stable cyclic graphs and trace theories (after Kaledin). On the other hand, Poincaré ∞-categories are a C2-refinement of stable ∞-categories that provide an adequate formalism for studying real and hermitian algebraic K-theory, which should be then well-approximated by the real cyclotomic trace. In this talk, we explain how to systematically provide Poincaré refinements of all the components of Nikolaus's approach to stable trace theories.
Let R be a commutative Noetherian ring and A a Noetherian R-algebra. In this talk, we study classification of torsion classes, torsion free classes and Serre subcategories of mod-A. In the case where A = R, such subcategories were classified by Gabriel, Takahashi and Stanley-Wang by using prime ideals of R. If R is a field, then A is a finite-dimensional algebra, and there are many studies of such subcategories relating with tilting theory. For a Noetherian algebra case, localization of A at a prime ideal of R plays an important role. We see that classification can be reduced to finite dimensional algebras. If A is commutative, our results cover cases of commutative rings.
In this mostly expository presentation, I will explain how certain combinatorial structures that arise in the representation theory of real reductive Lie groups can be used to solve several longstanding problems in classical invariant theory. Specifically, I will outline how to explicitly describe syzygies, Hilbert series, and linear bases of modules of covariants of several vectors and co-vectors.
The goal is to explain the title of the talk, and some consequences that flow from that property of the stable module category, having to do with locally dualizable objects.
Simple Lie superalgebras over complex numbers are of two types: Classical type and Cartan type. In our earlier joint work with Christodoulopoulou and Wiesner, we have defined Whittaker modules for Lie superalgebras and proved some important properties of these modules. Recently I was able to extend some of these results to Cartan-type Lie superalgebras. In this talk, I will briefly summarize these results.
The derived category of a commutative local noetherian ring and the module category of a modular group algebra are tensor triangulated categories. A dualizable object in such a category is one that has a dual that is compatible with the tensor structure. The question that we address in this paper is whether the subcategory dualizable objects in certain co-local subcategories is the idempotent closure of image of the compact objects under the local cohomology functor associated to the subcategory. In this lecture, I will try to explain what all of these words mean, why one might care about such a question and how we get a negative answer is certain cases.
Let G be an infinite discrete group. Finite dimensional unitary representations of G are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.
Recall that a noetherian ring R is regular if every finitely generated R-module has finite projective dimension. In a paper from 2009, Iacob and Iyengar characterize the regularity of R in terms of properties of (unbounded) R-complexes. Their proofs build on results of Jorgensen, Krause, and Neeman on compact generation of the homotopy categories of complexes of projective/injective/flat modules. In the commutative case, these results can be obtained with derived category methods in local algebra. I will illustrate how this is done by proving that the following conditions are equivalent for a commutative noetherian ring R:
1) R is regular.
2) Every complex of finitely generated projective R-modules is semi-projective.
3) Every complex of projective R-modules is semi-projective.
4) Every acyclic complex of projective R-modules is contractible.
The second condition is new, compared to the 2009 results, and relating it to the regularity of R is the novel part of the proof. This argument also plays a central role in the new proof of the corresponding results for complexes of injective modules and complexes of flat modules.
Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric group. These algebras enjoy distinguished bases whose product can be described on graph-theoretic diagrams. We extend this story to exterior powers, leading to the introduction of the mixed multiset partition algebra and a generalization of RSK that links the algebra’s graph-theoretic basis to a tableau basis for its irreducible representations.
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